Design of electromagnetic metasurface using two dimensional crystal nets

Metasurfaces are of great interest as they exhibit unique electromagnetic properties. Currently, metasurface design focuses on generating new meta-atoms and their combinations. Here a topological database, reticular chemistry structure resource (RCSR), is introduced to bring a new dimension and more possibilities for metasurface design. RCSR has over 200 two-dimensional crystal nets, among which 72 are identified as suitable for metasurface design. Using a simple metallic cross as the metaatom, 72 metasurfaces are constructed from the atom positions and lattice vectors of the crystal nets templates. The transmission curves of all the metasurfaces are calculated using the finite-difference time-domain method. The calculated transmission curves have good diversity, showing that the crystal nets approach is a new engineering dimension for metasurface design. Three clusters are found for the calculated curves using the K-means algorithm and principal component analysis. The structure–property relationship between metasurface topology and transmission curve is investigated, but no simple descriptor has been found, indicating that further work is still needed. The crystal net design approach developed in this work can be extended to three-dimensional design and other types of metamaterials like mechanical materials.

Some mistakes were corrected, and the files were converted to JSON format (see "Data Availability" section for the download link). There are 200 two-dimensional crystal nets in the original RCSR dataset, each has a net name, and several representative nets are shown in Fig. 1a. As the RCSR nets have different sizes, the nets in Fig. 1a were purposely selected in order to form a "square shape" for better visualization, and to give an idea that what the nets in RCSR look like. The names of the individual RCSR net are shown next to the structures. Here we briefly described how the crystal nets are named: the 3-letter net names, i.e., RCSR symbols, mostly inherited from zeolite framework type and mineral names, for example, "dia" means diamond structure. In some names, there is a hyphen, and the letter after the hyphen describes how the nets are derived from a simple common net (whose name only has three letters), for example -d refers to the dual net. More details can be found in the original RCSR paper 12,17 . Each RCSR net has two lattice parameters a and b, and an example is shown in Fig. 1b. RCSR net for metamaterials design. As RCSR is purposely designed for atomic structures, some of the crystal nets might not be suitable to be used as templates for metasurface design. Two criteria were applied to select suitable crystal nets. The first one is based on crystal symmetry. There are five two-dimensional Bravais lattice types (oblique, rectangular, centered rectangular, hexagonal and square). For simplicity with losing generality, only rectangular and square crystal nets are considered. The second criterion is based on vertex distance and metaatom size. In some crystal nets, the vertex is too close (i.e., the distance between two metaatoms is smaller than metaatom size), so the metaatoms centered in these vertices are overlapped. These crystal nets were removed. After applying the two criteria, only 72 crystal nets were kept for further investigation. The crystal nets in Fig. 1a all belong to these 72 selected nets.
To make a direct comparison of metasurfaces using different crystal nets, the lattice parameters of the metasurfaces unit cells are normalized based on metaatom "density" on the substrate. The reference structure is a simple square lattice (shown in net name sql in Fig. 1b). The lattice parameters of this structure are set to 8 mm, so the substrate area is 8 × 8 = 64 mm 2 . There is only one metaatom in the metasurface unit cell, so the metaatom "density" is 1 per 64 mm 2 . All other 71 metasurfaces are normalized to make their metaatom density equal to this value. This is done by changing the in-plane lattice parameters a and b of the unit cell, while keeping the a/b ratio the same as in the original crystal nets.

Electromagnetic simulation of RCSR-based metamaterials.
Greek crosses were used as metaatoms in the metasurfaces. The length of the Greek cross (4 mm) is the same in all 72 metasurfaces. Cross-type metasurfaces or frequency selective surfaces (FSSs) have been thoroughly investigated for decades 10 , here it is chosen www.nature.com/scientificreports/ due to its simplicity which facilitates the following analysis. The length of the Greek cross is carefully chosen by considering the second criterion in the RCSR net selection process (detailed in the previous part of "RCSR net for metamaterials design"): if the value of the length is too small, the whole structure cannot function properly as a frequency selective unit; if the value is too large, there will be too much overlap between the neighbor metaatoms and hence there will be only a few RCSR nets could be selected. As the metaatom density is fixed to "1 per 64 mm 2 ", we trialed several values and found that 4 mm makes a reasonable balance between the dilemma described above. The electromagnetic transmission curves were calculated using MEEP (MIT Electromagnetic Equation Propagation) 18 , an open-source software package for electromagnetics simulation via the finite-difference time-domain (FDTD) method. The simulation unit is shown in Fig. 1c. For simplification without losing generality, only the transverse electric (TE) component of the incident plane wave perpendicular to the metasurface and substrate is considered. The frequency range is set to 10-30 GHz, considering the corresponding wavelength with respect to the unit cell size of the metasurface.
Data and structure analysis. The calculated transmission curves were analyzed using Scikit-learn.
K-means algorithm was performed to cluster the structures using the transmission curves. n_cluster = 3 and default values were used. Principal component analysis (PCA) was used to calculate the coordinates for visualization. Euclidean distance was used in both functions as default.
Structural analysis is conducted using pymatgen package 19 , an open-source Python library for materials analysis. The metasurfaces were first converted to 3-dimensional atomic structure by adding a third dimension, i.e. a vacuum layer, which is suitable for pymatgen analysis (see source code for more details). Space group numbers were obtained using SpacegroupAnalyzer module. Inequivalent atoms were identified using get_symmetry_dataset method in SpacegroupAnalyzer module. Metaatom pair distances were calculated using distance_matrix function in the Structure class. The radial distribution function of the metasurface structure was calculated using RadialDistributionFunction implemented in Matminer 20 . Figure 2 shows the FDTD calculated transmission curves for 72 metasurface structures, and the results are aligned according to the alpha-beta order of crystal net names. The results of sql (right-most, second-last row) can be treated as a baseline as the sql (square lattice) structure is generally used for most reported metasurface. To give a quantitative comparison between the curves, clustering algorithms will be used to identify similar curves. Figure 3a shows the K-means (K is the number of clusters) clustering results of 72 transmission curves. Three K values (2, 3 and 4) were tested, and K = 3 gives the best visualization results, as shown in Fig. 3a. This also agrees with the visual intuition when looking at Fig. 2. The 3 clusters agree with the previous analysis for Fig. 2: the orange, blue or green clusters have transmission curves similar to sql, bew or cph, respectively. Figure 3b shows clusters of metasurface structures. To calculate the similarity between two metasurfaces, radial distribution function (RDF) is introduced, which is a metric widely used in materials science to measure similarities between different atomic structures 21 . RDF describes how the density of surrounding matter varies as a function of the distance from a point, and it is determined by calculating the distance between all particle pairs and binning them into a histogram. Then the histograms can be fed into K-means and PCA for clustering and visualization using Euclidean distance. As clearly shown in Fig. 3b, the metasurfaces are clustered into three groups using RDFs. Some representative structures in the groups are: sql for the blue group, where metaatoms are uniformly distributed in the metasurface; krm for the orange group, where metaatoms sit in different rows in a long lattice structure; cqk for the green group, where there is a large void inside the lattice. However, the correlations between clustering results of metasurface structures and transmission curves are weak. In other words, the metasurface in the same structural groups does not necessarily have similar transmission curves. Next, we will investigate whether there is some simple relationship between metasurface structures and transmission curves.

Results
To further explore the dependence of transmission curves on crystal net structural features, Fig. 4 plots the integral of the transmission curves (i.e. the area under the curve) against several crystal features typically used in atomic scale materials science. The features are (1) number of atoms: this is the number of crosses in each metasurface, as shown in Fig. 1a and insets of Fig. 2. As the "meta-atom density" are the same for all the metasurfaces (see "Method" Section for details), the number of atoms is also proportional to unit cell size. (2) Number of inequivalent atoms: in one structure, atoms are equivalent if they overlap to each other after a set of symmetrical operations. For example, in "cpa" structure, the four atoms at the corners are equivalent to each other, as by rotating 90 degree they overlap to each other. Hence "cpa" has 2 inequivalent atoms (while the number of atoms is 5), one in the center and the other at the corner. (3) Pair distance standard deviation: the distance of each pair of cross in one metasurface is calculated, and all the distance values form a "distance matrix" upon which the standard deviation is calculated. This is a measure of the "uniformness" of structure: smaller standard deviation value means the metasurface is more uniform. (4) Space group number: a higher space group number generally means that the metasurface is more symmetric. It is worth noting that 3D space group numbers are used here www.nature.com/scientificreports/ instead of 2D wall-paper group numbers for simplification, which will not affect the order, i.e., small 3D group number also means small 2D group number. As shown in Fig. 4, no strong dependence is found for all four structural features. This is not surprising as the "structure-property relationship" between metasurface structure and transmission curves can be complex. Although the four features/descriptors considered here are typical and widely used in materials science 22 , they might not be suitable to describe this relationship, as indicated by Fig. 4. However, it is still possible that suitable descriptor(s) for this relationship can be found, probably by applying feature selection techniques to millions of features. Such an approach has been successful in solid-state materials 23 , while for metamaterial further work is still needed.

Discussion
Reticular chemistry structure resource (RCSR) is introduced to electromagnetic metasurface design, and transmission curves of 72 metasurfaces constructed using RCSR crystal nets as templates are calculated. The calculated transmission curves are analyzed using K-means clustering and principal component analysis (PCA) for visualization. It is found that the transmission curves can be clustered into three major types. The dependence of transmission curves on metasurface structural features is also investigated. No simple relationship has been found indicating that the structure-property relationship between metasurface and transmission curve can be complex. The calculated transmission curves and the analysis provide basic and necessary information for further applying the topological net to metamaterials design. The main contribution of this work is providing a systematic way to investigate the new degree of freedom of "lattice structures" for metasurface design. From a "metamaterial data-driven design" point of view, the current work offers a dataset of metasurface structures and calculated transmission curves, which could be a reference for further metamaterial design work using this approach and can be used to build machine learning models.  www.nature.com/scientificreports/ From a practical point of view, the approach introduced in this work provides an alternative way (rather than changing the metaatoms) to modify the transmission curve. The above point can be further demonstrated by considering an application. In Fig. 2, all the transmission curves have a main valley around 22 GHz, and this feature makes them potential band-stop filters. It is worth noting that the Greek cross is not a state-of-the-art metaatom for band-stop filters, and it was chosen to facilitate high-throughput electromagnetic calculations and to ease the data analysis. Therefore, the purpose here is not to select the best metasurface as a band-stop filter but only to demonstrate the usefulness of the approach. Take sql (the square lattice) and sdq as examples: compared to sql, the metaatoms in sdq are shifted, and hence the position of the valley moves right-forward. Another example is comparing more complex patterns of metaatoms arrangement (e.g. cqm) to sql, the additional valley is introduced. These indicate that RCSR effectively changes the electromagnetic properties of the metasurfaces. Furthermore, if more suitable metaatoms are used, RCSR-based metasurfaces are expected to give better properties for specific applications, for which further work is needed.
Finally, RCSR 2D crystal nets only contain a small fraction of all possible 2D topological lattice structures, and there is a continuous effect to create new 2D nets. Recent progress is the development of a topology-based structure generator 24 implemented in USPEX package 25 , and this has been used in 2D nanomaterials design 26 . This also offers new opportunities for metasurface and metamaterial design to achieve desired properties.